At first, we investigate the effect of purity of quantum state on EPR
steering. The purity of a quantum state \(\varrho \)  is \(\mu \left( \varrho \right) =\mathbf {Tr}\varrho ^{2}\)  [1]}, which varies in the range
of \(\frac{1}{D}\le \mu \le 1\) , where \(D=\dim \mathcal {H}\)  for a given
Hilbert space \(\mathcal {H}\) . The minimum is reached by the totally random
mixture, the upper bound is saturated by pure states. In the limit of CV
systems (\(D\rightarrow \infty \)  ), the minimum purity tends to zero. The
purity of any N-mode Gaussian state is only function of the symplectic
eigenvalues \(\upsilon _{i}\)  of covariance matrix \(\sigma \)  [2]},
which is given by
\(\mu \left( \varrho \right) =\frac{1}{\prod _{i}\upsilon _{i}}=\frac{1}{\sqrt{\det \sigma }}.\) 
